How can I get the world grid and backdrop grid to be visible in a render?
Unfortunately, this is not possible using the standard render methods.

However, one way you can add it is by capturing an image of the grid as a screen shot from top orthographic view after hiding all visible objects. The grey background can then be removed using an image editing program and added to the scene as so:

Grid image (you are free to use this one if you like):


Then create a plane exactly the size of this image (6x6) and UV unwrap the plane

Cycles:

Set the material to the following node setup:


This makes the material only visible to the camera and transparent where there is no grid.

This can then be used in your renders :)

Blender internal:

For blender render simply enable Z-transparency in the materials and set the alpha to 0. Also enable shadeless and disable cast shadow.




Then select the image in the textures tab. Then ensure that the texture affects the alpha of the material by turning up alpha (just under color) to 1.



If the grid is too thick you can scroll up to image sampling, disable MIP map and lower the filter size. This should make the grid thinner.

You can now use it in your renders :)